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Properties

Label 2-2700-9.7-c1-0-16
Degree $2$
Conductor $2700$
Sign $-0.684 + 0.728i$
Analytic cond. $21.5596$
Root an. cond. $4.64323$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.70 − 2.94i)7-s + (−2.20 + 3.81i)11-s + (−1.03 − 1.78i)13-s + 1.40·17-s − 6.35·19-s + (0.0539 + 0.0933i)23-s + (4.54 − 7.86i)29-s + (−1.53 − 2.65i)31-s + 1.95·37-s + (−4.34 − 7.53i)41-s + (3.56 − 6.17i)43-s + (−3.74 + 6.48i)47-s + (−2.28 − 3.95i)49-s − 13.0·53-s + (−6.58 − 11.4i)59-s + ⋯
L(s)  = 1  + (0.642 − 1.11i)7-s + (−0.663 + 1.14i)11-s + (−0.286 − 0.495i)13-s + 0.339·17-s − 1.45·19-s + (0.0112 + 0.0194i)23-s + (0.843 − 1.46i)29-s + (−0.275 − 0.476i)31-s + 0.321·37-s + (−0.679 − 1.17i)41-s + (0.543 − 0.941i)43-s + (−0.545 + 0.945i)47-s + (−0.326 − 0.565i)49-s − 1.79·53-s + (−0.857 − 1.48i)59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.684 + 0.728i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.684 + 0.728i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2700\)    =    \(2^{2} \cdot 3^{3} \cdot 5^{2}\)
Sign: $-0.684 + 0.728i$
Analytic conductor: \(21.5596\)
Root analytic conductor: \(4.64323\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2700} (1801, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2700,\ (\ :1/2),\ -0.684 + 0.728i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9524443844\)
\(L(\frac12)\) \(\approx\) \(0.9524443844\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 + (-1.70 + 2.94i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (2.20 - 3.81i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.03 + 1.78i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 - 1.40T + 17T^{2} \)
19 \( 1 + 6.35T + 19T^{2} \)
23 \( 1 + (-0.0539 - 0.0933i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-4.54 + 7.86i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (1.53 + 2.65i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 1.95T + 37T^{2} \)
41 \( 1 + (4.34 + 7.53i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.56 + 6.17i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (3.74 - 6.48i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 13.0T + 53T^{2} \)
59 \( 1 + (6.58 + 11.4i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.862 + 1.49i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-6.18 - 10.7i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 7.50T + 71T^{2} \)
73 \( 1 + 5.42T + 73T^{2} \)
79 \( 1 + (4.71 - 8.17i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-4.48 + 7.77i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 4.01T + 89T^{2} \)
97 \( 1 + (1.08 - 1.88i)T + (-48.5 - 84.0i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.241854221082822611574291790361, −7.87903353508671643579487573454, −7.14988030474113520220804235230, −6.37178095313207016079511814588, −5.30313902338032194816813287620, −4.51774111676448274061594681834, −3.98428804014244302737861972707, −2.64176876165796587504933622915, −1.72984685500094716152526911333, −0.29567899488305437357988995353, 1.47878245339818417351268734893, 2.53762356528208536103592285304, 3.30900445827905201509081723525, 4.61351023994413010931123052095, 5.18675809127583025176545779913, 6.05396392683782687043153081878, 6.67892095902479132541478197391, 7.85612322115368534015045685951, 8.430027507360631288710104191339, 8.888096712624296420966371227212

Graph of the $Z$-function along the critical line